Archive for March, 2010

The Axiom of Choice states that given a family of non-empty sets Xi where i ranges over an index set I then there is a function, f, called the choice function, defined on I such that f(i) Î Xi for all i. For a finite number of sets the existence of the choice function follows by induction. It turns out that for an infinite number of sets one cannot deduce the existence of the choice function from the usual rules for building sets.

When the choice function is used in a proof then that part of the proof is nonconstructive.

The proof of the Banach-Tarski Paradox requires the axiom of choice. It shows that there is a way of dividing up the unit sphere into a finite number of subsets and then recombining these subsets, using rotations, reflections and translations, to form two solid unit spheres. The proof is nonconstructive. It can be shown that some of these sets are not measurable and this explains the apparent disconnect with the volume aspect of the paradox.

Two aspects of the axiom of choice are: the well-ordering principle and Zorn’s lemma. The well ordering principle states that every set can be well ordered. Zorn’s lemma states that under certain circumstances maximal elements exist. Zorn’s lemma can be used to show that every vector space has a basis. These two forms are equivalent to the axiom of choice.

In formal set theory how the axiom of choice is related to the other set theory axioms is studied.

In a two player game the game is called determined of one of the two players has a winning strategy. Using the axiom of choice on can construct nondetermined games.

The axiom of determinacy states that all games are determined. It contradicts the axiom of choice. The axiom of determinacy can be added to the Zermelo-fraenkel axioms without choice. It implies that many sets of reals have good properties such as being Lebesgue measurable. Variants of the axiom of determinacy are connected with the theory of large cardinals.

Banach spaces will be considered under the normed spaces and Banach spaces category.

Conditional probability is conditioned on some event. P(A|B) = P(A ,B)/P(B). A simple argument gives Bayes Theorem: P(A|B) = P(B|A)P(A)/P(B). This gives the conditional probability of A given B in terms of the conditional probability of B given A.

In statistics random data may be given by an unknown probability distribution.

If a and b are irrational numbers then can ab be rational? A proof of this is as follows. Let x = √2√2. Now either x is rational or it is irrational. If x is rational then we have irrational numbers, a and b, where ab is rational, namely √2√2. If x is irrational then we can have irrational numbers a = √2 and b = √2√2 with ab = (√2√2)√2 = √22 = 2 and hence ab = 2 is rational. This is called a nonconstructive proof. Because in the end even though we proved the proposition in which we were interested we do not know which a and b satisfy the proposition. In this proof we have used the law of the excluded middle.

The axiom of choice can be used to build sets out of other sets. The power set axiom says that given a set A we can form the set of all subsets of A. The axiom of comprehension is that for any set A and property p we can form the set of all elements of A that satisfy p. The axiom of choice is similar and says that we can make an arbitrary number of unspecified choices in order to form a new set.

The axiom of choice is often so natural that we may not realize that it is being used. One example is in the proof that the union of a countable number of sets of a countable number of objects is itself countable. The axiom of choice is used in the direct proof of this and it is a nonconstructive proof.

Another example is in graph theory to prove that an infinite disjoint union of even cycles is bipartite. A bipartite graph is one where the vertices of the graph can be split into two classes X and Y in such a way that no two vertices in the same class are connected by an edge (of the original graph).

Brouwer published two papers in 1918 and 1919 in German on “intuitionistic set theory”. From 1907 Brouwer rejected the principle of the excluded middle (PEM). PEM is the logical principle that the statement p .v p must always be true. The intuitionist view is that one can only state p .q when one can give either a constructive proof of p or a constructive proof of q.

A constructivist has a different view of what truth is. To say that a proposition is true simply means we can prove it by in accordance with the stringent methods they utilize. In this sense mathematics is not timeless. What is not true today may be true tomorrow.

Brouwer utilized concepts he called species, spread and choice sequence. A species is a set that has been defined by a characteristic property. A choice sequence may change over time. A spread has choice sequences as its elements. It is like a law that regulates how choice sequences are constructed. Brouwer has a concept of the continuum that is like that of Aristotle.

Brouwer defined a function to be the assignment of values to the elements of a spread. For Brouwer the function f(x) = 0 for x < 0 and =1 otherwise is not a well defined function.

The rejection of PEM also has the effect that intuitionistic negation and differs from classical negation. And intuitionistic arithmetic is different from classical arithmetic.

Godel and Gentzen were able to establish a correspondence between classical arithmetic and intuitionistic arithmetic.

In the 1920’s it became more and more clear that intuitionistic analysis was extremely complicated and foreign. Weyl initially supported intuitionism but eventually abandoned it. The Hilbert program approach suggested another way of rehabilitating classical mathematics.

Hilbert’s program, 1928, was an attempt to “eliminate from the world once and for all the skeptical doubts” of the acceptability of the classical ideas of mathematics. “The main goal was to establish, by means of syntactic consistency proofs, the logical acceptability of the principles and modes of inference of modern mathematics. Axiomatics, logic and formalization made it possible to study mathematical theories from a purely mathematical standpoint (hence the name metamathematics), and Hilbert hoped to establish the consistency of the theories by employing very weak means.” Hilbert “hoped to answer all the criticism of Weyl and Brower, and thereby justify set theory, the classical theory of real numbers, classical analysis, and of course classical logic with its PEM (the basis for indirect proofs by reducto ad absurdum).”

Hilbert realized that when theories are formalized any proof becomes a finite combinatorial object.

The theroies were also hoped to be complete in the sense that they would allow the derivation of all the relevant results. However, Gödel showed this to be impossible.

Hilbert’s theory proceeded gradually from weak theories to stronger theories. The word formalism to describe this theory came from formalizing each mathematical theory and formally studying its proof structure. Hilbert himself understood and valued the informal mathematical statements as well as the formal ones.

Hilbert and Brouwer were at odds and Hilbert eventually had Brouwer removed from the editorial board of Mathematische Annalen. Albert Einstein was slightly involved in this scuffle. Brouwer stopped being involved and the crisis slowly abated.

In 1931 Gödel proved that systems like axiomatic set theory and Dedekind-Peano arithmetic are incomplete. That is there exist propositions in the system such that neither the proposition or the negative of the proposition is provable. This is published in Monatshefte Fr Mathematik and Physik.

Gödel’s second theorem showed that it is impossible to establish the consistency of the systems that the first theorem dealt with by any proof that can be codified within them. Mathematics has continued to develop in light of Gödel’s results.

Around 1896 Cantor discovered the Burali-Forti paradox and the Cantor paradox. In the Burali-Forti paradox the assumption that all transfinite ordinals form a set leads, by Cantors results, to the result that there is an ordinal that is less than itself. A similar situation holds for cardinals and is called the Cantor paradox.

The comprehension principle is that given any well defined logical or mathematical property then there exists a set of all objects satisfying that property. In symbols this is given x there exists a set {x|p(x)}, where p represents the logical or mathematical property. Around 1901 Zermelo and Russel discovered a very elementary contradiction. The Zermelo-Russell paradox shows the comprehension principle is contradictory. Using the comprehension principle with the very basic property that p(x) = x Ï x we are led to a contradiction. If R = {x|p(x)} then if R Î R by definition we have R Ï R. Similarly, if R Ï R then, again by definition, R Î R. Thus we have a contradiction from some very basic principles.

In addition to these logical paradoxes there are many other semantic paradoxes formulated by Russell, Richard, Konig, Grelling and others.

These paradoxes caused a turmoil in mathematics. They led Hilbert to claim that to claim a set of mathematical objects exists is tantamount to proving that the corresponding axiom system is consistent or free of contradictions.

In 1903 Poincare used the paradoxes in the books of Frege and Russell to criticise logicism and formalism. He coined an important notion called predicativity. He claimed impredictive definitions should be avoided in mathematics. A definition is impredictive when it introduces an element by reference to a totality that already contains the element. Poincare found examples of impredicative procedures in all of the paradoxes he studied.

The successor function is s(N). Dedekind used this function and impredicativity to define the set of natural numbers. There was no problem or paradox here.

Richard’s paradox introduces definable real numbers. There must be a countable number of these. This leads to a paradox by using an argument similar to Cantor’s diagonal process used to prove that R is not countable.

In the predicativity approach all mathematical objects beyond the natural numbers must be introduced by explicit definitions. The definition must be predicative in the sense of referring only to totalities that have already been established before the object one is defining. Russell and Weyl accepted and developed this idea.

Zermelo pointed to cases where impredicativity was used without problems, including: Dedekind’s defining the set of natural numbers, Cauchy’s proof of the fundamental theorem of algebra and the least upper bound in real analysis. Zermelo said these definitions are innocuous in the sense that the object being defined is not created by these definitions it is merely singled out.

Poincare’s idea of abolishing impredicative definitions was utilized by Russell. He developed the theory of types which require members of sets to be of a similar type. This became the basis for the Principia Mathematica where whitehead and Russell 1910 -1913 developed a foundation of mathematics.

There were further studies of type theory by Chwistek, Ramsey, Russell, Weyl and recently by Feferman (1998). This approach does not fit neatly into the triad of logicism, formalism and intuitionism.

The “axiom of choice”, AC, is the principle that given any infinite family of disjoint nonempty sets, there is a set, known as a choice set, that contains exactly one element from each set in the family. Critics Borel, Baire and Lebesgue had all relied on AC to prove theorems in analysis. The axiom was suggested by Erhard Schmidt a student of Hilbert.

With Zermelo’s proof of the well ordering theorem he was led to develop the axiom system using the ideas of Dedekind and Cantor. With additions by Fraenkel, Von Neumann, Weyl and Skolem the axiom system became the one we know today. The system is called the ZFC system (for Zermelo-Fraenkel with choice). This codifies the key traits of modern mathematical methodology. It embodies the 4 principle of Hilbert and it or the von Neumann-Bernays-Godel version is the system that most mathematicians regard as the working foundation for their discipline.

The acceptance of impredicative and existential methodology became known as “Platonism”. Brouwer started to develop his ideas in 1905 and in his thesis in 1907. In intuitionism individual consciousness is the one and only source of knowledge. Brouwer contributed the fixed point theorem to topology. By the end of World War I he began publishing his foundational ideas. He helped to create the famous crisis. He established a distinction between formalism and intuitionism.

In 1921 Weyl published a paper in Mathematische Zeitschrift espousing intuitionism. Hilbert answered this criticizing Weyl and Brouwer. These ideas spurred Brouwer to a new theory of the continuum which enticed Weyl and and brought him to announce the coming of a new age.

The crisis in the foundation of mathematics is related to the three viewpoints of “logicism”, “formalism”, and “intuitionism” and the status of mathematical knowledge in light of Godel’s incompleteness results. This crisis is often considered to be localized around 1920 as a debate between the classical mathematics advocated by Hilbert and his critics led by Brouwer. There are aspects of this crisis which have been ongoing over a longer period of time.

“Logicism” is the thesis that the basic concepts of mathematics are definable by means of logical notions and the key principles of mathematics are deducible from logical principles alone. Hilbert endorsed this view starting around 1899. This view was consistent with that of Dedekind, Gauss and Dirichlet. This view was challenged by Riemann, Dedekind, Cantor, Hilbert and others. Opposition to this new direction came from Kronecker and Weierstrass.

The most characteristic traits of this modern approach were: i. acceptance of an arbitrary function proposed by Dirichlet, ii. infinite sets and the higher infinite, iii. put thoughts in place of calculations and a concentration on axiomatic structures and iv. purely existential methods of proof.

Dedekind utilized these traits in algebraic number theory with his set definition of number fields and ideals and in proving the fundamental theroem of unique factorization. Dedekind was able to prove that in any ring of algebraic integers ideals possess a unique decomposition into prime ideals.

Kronecker complained that these proofs were purely existential not allowing one to calculate, for example, the relevant divisors or ideals.

Dedekind and Riemann had ideas and methods that were formulated on general concepts rather than on formulas and calculations.

Weierstrass analytic or holomorphic functions as as power series of a certain form. He used analytic continuation to connect these with one another. Riemann defined a function that satisfies the Cauchy-Riemann differentiability conditions. Weierstrass offered examples of continuous functions that are nowhere differentiable.

The “conceptual approach” in 19-th century mathematics was that of Dirichlet’s abstract idea of a function as an arbitrary way of associating with each x some y = f(x). This was just the right framework to define and analyze general concepts such as continuity and integration.

The Bolzano-Weierstrass theorem which states that an infinite bounded set of real numbers has an accumulation point. Kronecker criticized this theorem because the accumulation point cannot be be constructed by elementary operations from the rational numbers.

Hilbert around 1890 had an existential proof of the basis theorem that ever ideal in a polynomial ring is finitely generated. Paul Gordan remarked humorously that this was theology and not mathematics.

Eventually the modern camp enrolled new members such as Hurwitz, Minkowski, Hilbert, Volterra, Peano and Hadamard and was defended by Klein.

After this followed the logical paradoxes discovered by Cantor, Russell, Zermelo, and others. The set theoretic paradoxes were arguments showing that assumptions that certain sets exist lead to contradictions. The semantic paradoxes showed difficulties with the notions of truth and definability. Around 1900 these paradoxes destroyed the attractive view of logicism.

Around 1900 Hilbert set down his famous list of problems. These included Cantor’s continuum problem and whether every set can be well ordered. These two problems and the axiom of choice were employed by Zermelo to show that the real numbers, R, can be well ordered.

In the twentieth century the actual mathematical proofs are seldom presented as fully formalized texts. They are arguments precise enough to convince a mathematician that it could be turned into a fully formalized text.

Hilbert and his collaborators around 1920 devised a notion of proof as a formalized and purely syntactic deductive argument. There soon came difficulties with this formalized proof idea. Around 1930 Godel came up with his incompleteness theorem. This showed that “mathematical truth” and “provability” are not the same thing. In any consistent sufficiently rich axiomatic system there are true mathematical statements that cannot be proved.

It was also proved that certain important mathematical statements are undecidable. In 1963 Paul Cohen established that the continuum hypothesis can be neither proved nor disproved in the usual axioms for set theory.

Some formal proofs are exceedingly long. The classification theorem for finite simple groups was worked out by large numbers of mathematicians in many parts and put together would reach about ten thousand pages. A proof so long that a single human being cannot check may be problematic to our conception of proof.

Sometimes the proofs are so complex that only a select few individuals are even capable of or qualified to evaluate the proof. Examples of this are the proof of Fermat’s last theorem and the Poincare conjecture. If someone claims to prove a theorem but no one wants to evaluate the proof then what is the status of the theorem?

With modern computers and probabilistic methods it is possible to prove theorems with a high probability of them being true. Are these proofs? Even though they may have less likely error than the long classification theorem for finite simple groups mentioned above!

In 1976 Kenneth Appel and Wolfgang Haken proved the four color theorem which required the use of a computer. This is an example of a computer assisted proof. The future may hold computer generated proofs.

Another issue is that some conjectures that appear to be true are the basis for much research under the assumption that they are true. How should the results of such research be viewed?

The formal notion of proof is as a string of symbols that obeys certain syntactical rules. This notion continues to provide an ideal model for the principles that underlie what most mathematicians see as an essence of their discipline.

Cauchy, Weierstrass, Cantor and Dedekind aimed at eliminating intuitive arguments and concepts instead using more elementary statements and definitions. The idea of pursuing an axiomatic basis of mathematical theories was pursued in the nineteenth century by George Peacock, Charles Babbage, John Herschel and Hermann Grassmann. Giuseppe Peano in 1889 “Postulates for the Natural Numbers” used an artificial language that would allow a completely formal treatment of mathematical proofs and applied an axiomatic approach to arithmetic. This did not lead Peano to a more formal conception of proof. Mario Pieri developed a theory to handle abstract formal theories.

At the end of the nineteenth century Hilbert published his “Grundlagen der Geometrie”. In this he synthesized and completed the prior trends of geometric research. He introduced a generalized analytic geometry. In these the coordinates may be from a variety of different number fields rather than just the real numbers. One of the purposes of the logical analysis was to avoid being misled by diagrams. His axioms for geometry were of five types: axioms of incidence, of order, of congruence, of parallels and of continuity. He also required the axioms be independent, consistent and simple.

Eliakim Moore in the beginning of the twentieth century turned the study of systems of postulates into a mathematical field in its own right.Felix Hausdorff in 1904 was among the first mathematicians to associate Hilbert’s work with a new formalistic view of geometry. Other scientists who reacted to Hilbert’s approach are: Frege, Boole, De Morgan, Grassmann, Charles S. Peirce and Ernst Schroder.

The inclusion of the understanding of logical quantifiers universal, , and existential, $, was very significant in a step toward a new formal conception of logic. Cauchy, Bolzano and Weierstrass

Fregein 1879 proposed a system in his work “Begriffsschrift”.There were similar ones proposed by Peano and Russel. Frege had a formal system with possibility of syntactic checking of any deduction. “It is the truth of axioms, asserted Frege, that certifies their consistency, rather than the other way around, as Hilbert suggested.”

The foundational research in geometry and analysis converged to create a new conception of mathematical proof. A mathematical proof is seen as a purely logical construct validated in purely syntactic terms, independently of any visualization through diagrams. This conception has dominated mathematics ever since.

In the eighteenth century Euler reformulated calculus and separated it from its geometrical roots. D’Alembert associated mathematical certainty with algebra. Lagrange in his work “Mechanique Analitique” expressed the view “One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course.”

Kant refers to “visualizable intuitions” or “anschauung” in German. He draws a distinction between a philosophical argument and a geometric proof. He claims the geometric proof utilizes the concrete concept of visualizable intuitions. To Kant a geometrical proof is constrained by logic but it is much more than just a purely logical analysis of the terms involved – it includes visualizable intuitions.

By the end of the nineteenth century the concept of a proof and its role in mathematics had become deeply transformed. In 1854 Riemann in Gottingen gave a talk on “On the Hypotheses Which Lie at the Foundations of Geometry”. In the 1830’s the ideas of Bolyai and Lobachevski on non-Euclidean geometry as well as related ideas of Gauss became known. In 1822 Jean Poncelet’s treatise on projective geometry opened foundational questions. Klein and Lie studied group theoretic ideas in the 1870’s. In 1882 Moritz Pasch had an influential treatise on projective geometry that handled many foundational issues. It also closed some of the gaps in Euclidean geometry. Pasch put a much greater emphasis on the pure logical structure of the proof. He still however utilized diagrams.

Khwarizmi in the eighth century included algebraization of mathematical thinking and Euclidean geometric proof to legitimize his mathematics. His Islamic book in translation entitled “The Compendious Book on Calculation by Completion and Balancing” is an example of these ideas. He includes variables along with numbers in his problems and solutions. The quadratic equation is broken up into six cases and is not yet treated as a singular entity. On translation an example of this is the problem “What is the square which combined with ten of its roots will give a sum total of 39?” This may now be written as: x2 + 10x = 39. The solution is written out more in recipe form. A justification is given by a geometric argument.

Cardano in the 1545 book “Ars Magna” presented a complete treatment of 3rd and 4th degree polynomials. Even though his reasoning is more abstract than that of Khwarizmi he still continued to utilize diagrams of Euclid-like geometric arguments.

In the seventeenth century Newton and Leibniz created and utilized infinitesimal calculus in their proofs. Fermat and Descartes further utilized these tools. More progressive ideas than geometrical proof were also utilized by Cavalieri, Roberval and Torricelli. In 1643 Torricelli used an example of computing the volume of a rotating hyperbola.

In 1637 Descarte in “La Geometrie” algebratized geometry by making an analogy between operations in arithmetic with lines and line segments in geometry. He included a unit length and was able to remove the dimensionality of values and thus do away with the requirement of homogeneity of variables.

It became possible to prove geometric facts with algebraic procedures. This idea reached full maturity by Viete in 1591. Newton (1707) was not a proponent of algebraic thinking. In his “Principia” he used his own calculus sparingly and only where necessary and barred algebra from his treatise entirely.

The development of the idea of proof seemed to occur around 500 B.C. with deductive argument. Thales of Miletus, mathematician, philosopher and scientist, is the first mathematician known by name. Euclid’s books the “Elements” were written around 300 B.C. and there were one of the most successful attempts to organize mathematical concepts. Mathematics stands out in the sciences in the unique way in which t relies on “proof”. There are other attempts of this kind including induction which are not dealt with in this present account.

There was an early reliance on geometric justifications. These later, with the development of non-Euclidean geometry and problems in the foundations of analysis, led to a fundamental change in mathematical orientation. Arithmetic and set theory eventually provided the certainty and clarity from which other disciplines drew their legitimacy and clarity.

The proofs in Euclid’s Elements all have six parts and an associated diagram. Some terms in the elements now have different meanings, for example, two figures are said to be equal if their areas are equal. The six parts of the proofs are: Protasis (enunciation), Ekthesis (setting out), Diorismos (construction), Kataskeue (construction), Apodeixis (proof) and Sumperasma (conclusion).

An example from the elements is the following IX.35 which reads, after translation by Sir Thomas Heath, as follows: “If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first , then, as the excess of the second is to the first, so will the excess of the last be to all those before it.” In today’s mathematics this may be written: (an+1 – a1) : (a1 + a2 + + an) = (a2 – a1) : a1.

There was an accompanying figure for this proposition IX.35 which included four line segments of different lengths with various points included on these segments. The proof refers to this diagram.

In Greek mathematical texts proofs sometimes relied on the “exhaustion method” which was based on a double contradiction. An example of this method is provided in Euclid’s proposition XII.2 of Euclid’s Elements. Other methods were sometimes used in proof including: synchronized motion of two lines and mechanical devices. However, the Euclid style of proof remained a model to be followed whenever possible.